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- PHAN HUY PHIJ • NGUYEN DOAN TUAN
BAI TAP
DAI SO TUYEN TINH
NHA XUAT BAN HAI HOC QUOC GIA HA NOI
- Chin trach nhiem xual bcin
doe:
Gicim NGUYEN VAN THOA
Tong bien Op: NGUYEN THIEF N GIAP
Bien tap: HUY CHU
DOAN 'MAN
NGOC QUYEN
Trinh bay Ilia: NGOC ANH
BAI TAP DAI sq TUYEN TINH
Ma s6: 01.249.0K.2002
In I .501) cudn, tai Xtiiing in NXI3 Giao thong van tai
S6 xuat ban: 49/ 171/CXS. S6 Inch ngang 39 KH/XB
In xong va Opt [Yu chi& CM/ I narn 2002.
- Lai NOI DAU
Mon Dai s$ tuygn tinh dude dua vao giang day a hau hat
cac trUnng dai hoc va cao dang nhtt 1a mot mon hoc cd se; can
thigt d@ tigp thu nhUng mon hoc khan. Nham cung cap them
mot tai lieu tham khao phut vu cho sinh vien nganh Toan vi
cac nganh Ki thuat, chting Col Bien soan cugn "BM tap Dai so
tuygn tinh". Cugn each dude chia lam ba chudng bao g6m
nhUng van d6 Cd ban cna Dal so tuygn tinh: Dinh thfic va ma
trail - Khong gian tuygn tinh, anh xa tuygn tinh, he phticing
trinh tuygn tinh - Dang than phttdng.
Trong mOi chudng chung toi trinh bay phan torn tat lY
thuyat, cac vi du, cac hal tap W giai va cugi mOi chudng c6 phan
hudng dan (HD) hoac dap s6 (DS). Cac vi du va bai tap &roc
chon be a mac an to trung binh den kh6, c6 nhUng bai tap
mang tinh 1± thuygt va nhUng bai tap ran luyen ki nang nham
gain sinh vien higu sau them mon lice.
Chung toi xin cam on Ban bien tap nha xugt ban Dai hoc
Qugc gia Ha Nei da Lao digt, kien de cugn sach som dude ra mat
ban doe.
Mac du chting tea da sa dung 'Lai lieu nay nhigu narn cho
sinh vien Toan Dal hoc Su pham Ha NOi va da co nhieu co gang
khi bier, soon, nhUng chat than con có khigm khuygt. Cluing
toi rat mong nhan dude nhUng y kin clang gap cna dee gia.
Ha N0i, thcing 3 !Lam 2001
NhOni bien soan
3
- rvikic LUC
7
Chubhg .1: DINH THOC - MA TRA:N
7
A - Tom tat ly thuyeet
7
§1. Phep th6
§ 2. Dinh thitc
10
§ 3. Ma tram
12
B - Vi dn
35
C - Bei tap
43
D HtiOng dein hoac clap so
Chudng 2. KHONG GIAN VECTO - ANH XA TUYEN TINH
57
PHUGNG TRINH TUYEN TINH
•
57
A - TOrn tat ly thuyeet
57
§1. Kh8ng gian vec to
61
§2. Anh xa tuyeen tinh
64
§ 3. He phydng trinh tuy6n tinh
67
§4. Can true caa tai ding cku
71
B Vi dtt
96
C - Biti tap
96
§1. 'thong gian vec to va anh xa tuyeen tinh
104
§2. He pinking trinh tuy6n tinh
106
§3. Cau tit cna melt tu thing calu
110
D. Illidng sign ho(tc clap s6
5
- §1. Khong gian vec td va anh xn tuyin tinh 11(
§ 2. He phudng trinh tuyeit tinh 12';
§3. Cau trite dm mot tg ang cau 12Z
Chtedng DANG TOAN PHUONG - KHONG GIAN VEC TO
OCLIT VA KHONG GIAN VEC TO UNITA 134
A. Tom Vitt 1t thuyeet 134
§1. Dang song tuy6n tinh aol xUng va dang town phuong 139
§ 2. Killing gian vec to gent 135
§3. Khong gian vec to Unita 142
B. Vi du 14E
C - Bai DM 174
D. Hitting dan hotic ditp so 179
Tai lieu them khan 192
6
- Chuang 1
DINH THUG - MA TRAN
A - TOM TAT Lt THUYET
§1. PHEP THE
n} len chinh no duet goi la
Met song anh o tit tap 11, 2,
met phep the bac n, ki hieu la
'1 2 3
a2 G
G 3
I
15 del a, = a(1), 0 2 = a( 2),..., a„ = a(n).
Tap cac phep the bac n yeti phep nhan anh xa lap thanh
met nhom, goi la nh6m del xeing bac n, ki hieu S. S6 cac Olen
t3 cua nhom S„ bang n! = 1, 2... n.
Khi n > 1, cap s6 j} (khong thu tv) dude pi IA met nghich
the cem a n6u s6 - j) (a, a) am. Phep the a &foe goi la than
-
ndeM s6 nghich thg. cim a chan, a &toe goi la phep the le n6u s6
nghich the ciaa a le.
1 neM s la phep the chan
Ki hieu sgna =
-1 net} a la phep th6 le
va sgna goi IA deu am, phep the a. Neu a vat la hai phOp the
= sgn(a) . sgn( ).
cling bac, thi sgn(a
Phep the a chicly goi IA met yang xich do dai k n6u c6 k s6 i„
coo = 1 2 , coo = i3, a(ic) = i1
• - • , i k doi mot khac nhau dr
7
- va a(i) = i vdi moi i x i„ i k . Vong )(felt do dttoc ki hieu IA
i k ). M9i phep th6 dau &tan tfch the thanh tfch nhung
yang xfch doe lap.
Met vOng xfch do dal 2 dude goi IA met chuygn trf. Vong
xfch ••• , i k ) phan tfch chive thanh tfch 0 1 ,
§ 2. DINH THUG
I. Gia sit K IA met trueng (trong cuan sich nay to din yau
xet K la &Ong s6thvc K hoac truang s6 phitc C). Ma tran kidu
(m, n) vdi cox phan tit troll twang IC la met bang chit nhat gfim
m hang, n cet cac phan tit K, i = 1,m, j = 1,n. Tap cac ma
tran kidu (m, n) chive kf hieu M(m, n, R). Ma trail vuong cap n
IA ma tran co n dong, n cot. Tap cac ma trail vu8ng cap n vdi cac
phan tit thuoc truong K ki hiOu IA Mat(n, K).
2. Cho ma tr4n A vuong cap n, A = (ad, i, j = 1, 2, ..., n.
Dinh thitc ciia ma tran A, kf hieu det A la met flan tit dm K
dude xac dinh nhu sau:
detA = zsgn(a)a mo)
Sn
E
3. Tinh eh& ceta Binh that
a) Neu dgi cho hai dong (hoac hai cot) nao do cim ma tram
A, thi dinh auk cim no ddi da:u.
b) N6u them veo met dong (hoac met cot) cim ma tran A
met to hdp tuygn tinh cim nhUng thing (hoac nhung khac,
thi dinh auk khong thay ddi.
8
- •
phan tfch thanh tong, thi
c) Ngu mot Bong (hay mot
dinh thitc dU9c phan tfch thanh tong hai dinh thfic, cv th6:
an,
an ail
f
a21 +a lci a2„
de
a,,, + an i
‘a n„
a ll
a ll a l; ...a 1 ,„
a21 ...a2 n
a 21 21 + de t
= det
—a 1111/ " S ' Ill " S IM /
a do
d) Cho A = (It o ) Mat(n, K), thi = b) = a ij &toe
E
goi la ma tran chuy6n vi cim A.
Ta co detA = detA t .
4. Cdch tinh dinh that
a) Cho ma tran A Mat(n, K). Kf hi'911 M i; la dinh that cua
E
ma trail alp (n-1) nhan dine bAng cach gach be clOng thU i, cot
thu j cut ma tram A vb. Aij = (-1)H M u clucic g9i la pha'n phu dai
s6cUa phgn to a ii cna ma trait A. Ta có CAC tong thtic:
O ngu i k
det A ngu i = k
ngu i x k
O
det A ngu i = k
Nhu fly detA = EamAki (k = 1, 2, ... n)
1=1
heat detA = Z a ik A ik
/=1
9
- CUT thac tit throe goi la cang thdc khai trim dinh tilde
theo (long hay theo cot.
b) Dinh 1ST Laplace
Cho ma Iran A = (a, J ) c Mat(n, K). Vo; rn6i bQ
1 s i,
-
aln
811 a19
aon
a99
a, 1
A=
arnn,
amt
amt
2. Cac phep todn tren Mat(m, n,
Cho A = (a y ), B= (b, j ) thuOc Mat(m. n, K)
Ta có:
a) Ma tran C = (cg) a do c y = a ii +
&toe goi la tong cua hai ma tran A va B va ki hien la A +B.
a do d i; = a ij -
Ma tran D= (d,,)
dude goi la hiOu cila ma trail A va B va Id hi'eu la A - B.
ma trail kA c8 cac phAn to la (ka ii ) duoc goi
b) Vdi k E At,
la tick cua ma tran A vdi ph&n td k cua trudng K.
A = (a ii ) c Mat(m, n, K) va
c) Neu
B = (bp( ) e Mat(n, p, K) thi
Mat(m, p, K) ma cac phAn tit &tele xac dinh INN
ma tran A . B E
a do
AB = (c, k ),
e ik = Zaijbjk
5=1
&toe goi litich caa hai ma tram B ye. A.
Vol A, B e Mat(n, K), to có det(AB) = detA. detB.
11
- d) Tap Mat(n, K) con ma tran yang cap n vdi phep toan
cOng lap thanh mot nhom giao hoan, con vdi phep Wan rang ma
tran va phep nhan ma trail lap thanh mat vanh khong giao
hodn, co don vi.
3. Hang ctia ma tran; Ma trim nghich ddo
Gig A Mat(m, n, K), ta dinh nghia hang ciat ma tran A
E
la cap cao nhgt cua dinh thric con khgc khong rut ra W ma tran
A. KM A E Mat(n, K) va hang A = n (ta cling dung ki hi3u hang
A la rang A) thi ma tran A goi la khong suy bign, khi do
detA * 0 va ton tai duy nhgt ma tran B thuOc M(n, K)
A.B = B.A = I„; d do I. lit ma trail don vi. Ma tran B &roc goi 11
ma tran nghich dgo cna ma tran A va ki hi3u la A'.
Gig su A= (A u )la ma trail plw hpp cim ma tran A = (ad,
A b la Olga Ow dal see mitt phgn ht a ii ; A t la ma tran chuya'n
vi cua A . Khi do:
At .
detA
B- VI DTI
Vi cla 1.1. Xac dinh clgu rim cac phep th6 saw
11 2 3 4 5
a) a
2 3 5 4 1
r1 2 3 n n+1 n+2 211 2n+1 31
b) 5= I
ll 4 7 ai-2 2 5
12
- Lai gidi
a) Phan tick) a thanh tich cac chuydn tri:
2 3 4 5)
= = (1 2 3 5) = (1, 5) (1, 3) (1, 2)
(2 3 5 4 1
1
(chi) 9 la pile)) nhtin cae chuydn tri dude thuc hign tii phai
sang trai nhu hap thanh cua the Anh xa).
Vay sgna = (-1) s = -1
Co the lam each khan: Cam nghich the cua a la (1, 5), (2, 5),
(3, 5), (4, 5), (3, 4).
Vay a có 5 nghich the nen sgna = -1.
b) Ta hay tinh sS nghich the cua boa)) vi (1, 4, 7... 3n-2, 2,
1 khong tham gia vao nghich the
4 tham gia yen 2 nghich the voi the s6 thing sau no.
7 tham gia van 4 nghich the.
3n - 2 tham gia vao 2(n - 1) nghich the voi the se dung sau no.
2 khong tham gia vao nghich the nao vdi the se dung sau no.
5 tham gia yen 1 nghich the voi the se dung sau n6.
S tham gia vao 2 nghich the vdi cae s6 dUng sau n6.
3n - 1 tham gia vim (n - 1) nghich the voi the s6 thing sau no.
Cae s6 3, 6, 9..., 3n khong tham gia vao nghich the nao voi
the s6 (hang sau
13
- Vay co tat ca 2 + 4 + 2(n-1) + 1 + 2... + (n 1) - 3(n -1)n
2
(n-1 )n
nghich the trong hoot vi da neu va do d6 sgn S = (-1) 2
Khi n = 4k hoac n = 4k + 1 thi sgn 5 = 1
con neu n = 4k + 2 ho4 n = 4k + 3 thi sgn = -1.
Vi du 1.2
_123
Cho phep th'e' f - en dgu la (-1) 1
fn
12 3
Hay gag dinh da"u dm:
a) 1 -1
2
b) g = (In
fn-) • n)
Lift( gidi:
a) Vi sgn f. sgn = sgn (f.
= sgn(Id) = 1 non
sgn (e 1 )= sgn (f) = (-1) k
nj
b) X4t phep the"a = 1 2
n -1 . 1
thi g = f. a
Do gay sgn g = sgnf . sgn a.
n(n-1)
2 nen sgn g = (_])k+C;',
Nhung sgn a = (-1)
14
- Vida 1.3
ChUng minh rang vier nhan mat phep th6 vdi ehuy6n trf
a clang
j) v6 ben trai Wring throng v6i viac dai cha car s6 i j ,
drah cna phep the . . Cling nhu vay, nhan mat phop th6 Nth ehuynn
trf (i, j) v6 been phai tunng during voi del eh?, I, j a dong tit 66a
phep th6.
Lo gidi
Gia sif a la phop th6 cho j) la phep chuy6n tri. Xet
truong hop nhan ben trai tile la f = (i, j). a.
(3 n
Gitisi2 a=
1 2
Theo d nh nghia (i, j) = (
9 2 nj
_ (1 2a n
ai )
Trunng hop nhan ben phai dude ant Wring fib
Vi dy 1.4. Cho f va g la hal phop th6cua n strtn nhien clAu tien.
a) Chung minh rang có the' cilia f va g bang khong qua (n - 1)
phop chuyan trf (nghia la ton tai k phop chuyan trf a l , cr 2 , ak ,
g = a k a k _,... a,. f).
k 5.11 - 1
b) Chling minh rang khOng tha giam bat s6 chuOn trf rah
trong cau a) titc la en the' chon f va g sao cho khong the dua f vd
g bang ft han n - 1 phop chuyan trf.
15
-
La gicii
a) Xet phep the g o f', phan tich g o e' thanh tfch cac vang
xich dOc lap T 1 , Tp .
e' = ...
go T 2 .T i
Neu kf hiOu m i nt do, dai cart yang 'dell T i thi
± m2 + =
rang mOt vong xfch (a l , a 2 , a m ) la mOt plop the
a cac s6 tv nhien Ui 1 den n sao cho a(a) = a 1+1 (i = m-1) va.
a(a„,) = a l , con a(l) = 1 nen 1 yen moi i = 1, .,., m. Vong xfch
(a l , a 2 , u„) goi la ce do, dai m.
Ta da hiet rad yang xfch do, dai m deu phan tich duo
thanh m - 1 chuyen trf. Vi vay g o e' phan tfch duo. thanh 'Lich
caa i(m i -1)= n -p = k phep chuyen trf.
i=1
Nhungpa. 1 n-1:115n-1
Nhu vay g o f -1 = a k (s, - chuyen tri)
TV do g = a k 0 ak.,, ... 0 0 f, kn-1
va cac a, la cac phep chuyen trf.
ri nj
b) Cho g = la phep the ddng nhA
0. 2 ...
23
1
on f =
(
. Ta se chUng to rang killing dua
Oa 1 2... n-1)
n
f ve g duo Wang it hdn n - 1 phep chuyen trf.
16
-
'
ta not rang la
WA met phep the" h = 1
1 2 n
2
i. De" rang neat nhgn vao ben trai
phan tit chinh quy ngu
cua h met chuygn Lri thi A:C . 1)111in tit chinh quy tang cling lAm la
met don vi. That way, ngu ngudc kg, cheng hen i, j la hai phan
tit kheng chinh quy cim h ma nob d6i dig h i voi h j ta Jai &tog
hai phign tu chinh guy (cum phep th6 m6i) th6 thi: h, < i. h, < j
nhang hj 2 i, h ; j vti 1Y.
Do f chi co met phAn tei chinh quy. ye g co n phan tV chinh
quy, vi vgy khong thg clua f vg g bring it hon n - 1 phep chuygn tri.
Vi du 1.5
Chung minh rang vdi mei so k (0
-
Nhan cot thu nhal ciia ma tran A vdi -k rdi cOng vac) cot
this k, ta dude:
1 -1 -2 ... -(n-1)
- (n - 2)
1 0 -1
det A = 1 0 0 .. - (n - 3)
1 0 0 0
Khai trio'n the() dung Ulu n, ta ea:
-1 -2 ... -(n -1)
-1
= (-1)" +1. (-1)"-'=1
Cdch 2.
Ta tha'y A= B. ca do
1 t1
1 0
vi C=
B=
1
11
ma detB =1, detC = 1 nen detA = detB. detC = 1.
Vi du 1.9. Hay tinh
cosa 1 0 0 0
1 2cosa 1
= 0 1 2cosa 0 0
2cosa 1
1 2cosa
0 0 0
21
- Lai gidi:
Khai trim dinh thac Lheo cot cub" to co
-
D„ = 2cosa . 17,1 _2
De thay D, = cosa.
1
cosa
= 2coi2 a - 1 = cos2a.
2cosa,
Gia sa D 1 = cosia \TM moi = 1, . k.
Ta có
2cosa .
Dk,I = Dk - Dk_ i
= 2.cosa . coska - cos(k -Da.
= (cos(k+Da + cos(k-1)x) - cos(k-1)a = cos(k+1)a.
Nhn vay D„ = cosna
Vi do 1.10
Hay Dull
1 0
+ a -9
0
1 e" +e 1
A„ = N x0
1
1
1 eP + e -(1)
0
a do the phan to tren &tang choo chinh bang nhau va band
e q) +e -9 ; the phan tit tren hai &tong xien Win nhat \TM (Mang
the() chinh bang 1, con the phAn ta khac bang 0.
22
- Khai trin theo cot tht nhEt, to c6:
A n = (e P +e - P)A n _ i:
e 21' - e -2(P
Nlinn xet rang 4 1 = 6 9 +C c =
-
36 - -39
6
e
e ro
((i
A2 =
e (P -
e (1+1)6 - e -(lrv1),p
, n - 1.
k - 1, 2,
Girt sit ,
AR -
e (-0 -e (P
Ta c6 An = (e c e - P)A n _ i -A n _,
(:+1)o - e -(n+1)4'
nip - e npe( n-1) n4) e
- e
=(eP +e w)e
—
ew e q) -
-(n+1),p
e (n+1)p -e
Nhu v 6}.:
. An =
e 1 -e
Vi du 1.11
Tinh:
an
a,
ll a1
an
a l: +h i a,
1
an
D = dot 1 a, +b.,
a1
a n +b n
a1 a.9 ...
23
- Lai gidi:
LAY ciOng dAu nhan vOi -1 r6i Ong vao cac thing con
1ai to có ngay D = b, 1) 2 ...b„.
,
Vi du 1.12
Cho da thric P(x)=x(x+1)...(x+n)
Hay tinh Binh thdc:
P(x) P(x +1) P(x + n)
P(x) P(x +1) ... P(x +n)
d=
P (n-1) (x) P th-1) (x+1) P th-1) (x + n)
P (n) (x +n)
P thl (x) P thl (x +1)
gidi:
Ta b6 sung de' dude ma Han dip (n+2):
P(x) P(x +1) P(x +n) 0
P(x) P4x+1) P4x+n) 0
D=
P ( n ) (x) Pthd(x +1) P 0P(x + n) 0
pg+0( x +n ) 1
P („+l) (x) 1301+1) (x +1)
RO rang det D = d
(x +n
Nhan dOng 1111 k cua ma trail D vdi dc-ix( 1) k-1 r6i
(k-1)!
Ong vao clang Hirt nhgt vgi tat ca k=2, .. n+2).
Khi do, phAn tii dung dau có clang:
poc + 0 + P (k) (x +0.(x +11.) k = n).
ok
k!
k=1
24
nguon tai.lieu . vn